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Algebra II – Unit 5 Numbers and Functions OVERVIEW Why is this important to learn? Numbers and Functions is expected to take approximately 4 weeks to complete during the first semester of the Algebra II course. This unit will be geared to secondary students desiring a credit in Algebra II. In order to be successful in this class, students should have successfully completed Algebra I. It is absolutely necessary to have the ability to relate one quantity to another in all academic areas and the function notation allows this to be accomplished. The ability to use function notation, manipulate, transform and perform mathematical operations on functions is a key element in the study of higher mathematics. In addition, this skill allows the student to relate two real world quantities in other academic areas like business and science. BIG IDEAS: The big ideas of this overview include: Determining the best method for solving problems requires critical thinking. Sometimes problems have more than one solution. Modeling real world situations requires a multitude of different types of functions. Different functions share common characteristics. Inverse operations are a key element in solving all types of functions. You can add, subtract, multiply or divide functions based on how you perform these operations for real numbers. You can represent functions in a variety of ways such as graphs, tables, or words. Each representation is particularly useful in certain situations. You can model a quantity that changes regularly over time by a certain percent. ESSENTIAL QUESTIONS: Questions that will help students process content include . . . What problems can we solve using piecewise defined functions? What do the key features of a graph mean in the context of a problem? What kind of practical problems can be solved using function? How does a function express a mathematical relationship between two related quantities? How can you model a quantity that changes by a percent over time? CONTENT AND SUBSTANCE What is the learning? DISTRICT STANDARDS: Understand solving equations as a process of reasoning and explain the reasoning (A-REI.1) Interpret functions that arise in applications in terms of the context (F-IF.2) Analyze functions using different representations (F-IF.3) Build a function that models a relationship between two quantities (F-BF.1) Build new functions from existing functions (F-BF.2) 1 LEARNING TARGET(S) To understand, students will need to know, understand and do . . . Vocabulary: Intercepts, intervals (increasing and decreasing), relative max/min, end behavior, piecewise function, explicit expression, recursive process, exponential growth and decay, invertible function. Declarative Knowledge: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. (F-IF.2A) Procedural Knowledge: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. (A-REI.1A) For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. (F-IF.2A) Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★ Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. (F-IF.3Aii) Write a function that describes a relationship between two quantities. ★ Determine an explicit expression, a recursive process, or steps for calculation from a context. (F-BF. 1Ai) Write a function that describes a relationship between two quantities. ★ Combine standard function types using arithmetic operations (F-BF. 1Aii) Write a function that describes a relationship between two quantities.★ Compose functions. (F-BF. 1Aiii) Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. Find inverse functions (F-BF.2Bi) Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. (+) Verify by composition that one function is the inverse of another. (F-BF.2Bii) Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. (F-BF.2Biii) Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. (+) Produce an invertible function from a non-invertible function by restricting the domain. (F-BF.2Biv) 2 ASSESSMENT EVIDENCE/CRITERIA How will students demonstrate their understanding and how will you know? The assessment is aligned to the common core learning targets and standards. Evidenced by the common assessment keys. A unit pre-assessment may be delivered informally. In addition, unit exams will give evidence of the level of student knowledge and ability to be successful on a common task. Teachers will administer independent summative assessments by unit and use the common district task for multiple units. Depth of knowledge of unit assessments will be determined by individual teachers; however, they will reflect the requirements of the common assessment as evidenced by the keys. A formal 4 tier mathematics scoring rubric will be used on common assessments throughout the district. Unit assessments will be scored by individual teachers. INSTRUCTIONAL PLANNING AND RESOURCES What learning experiences will help all students meet the identified learning targets? Components may include: Resources – texts, lab materials, math tools, technology, etc. Practice lessons White board practice Pre-assessments Vocabulary work Word wall Student groupings Activities Independent work time Differentiated instruction 3